We describe the new field of the mathematical analysis of deep learning. This field emerged
around a list of research questions that were not answered within the classical framework of …

This paper analyzes the rapid and unexpected rise of deep learning within Artificial
Intelligence and its applications. It tackles the possible reasons for this remarkable success …

In this paper, we prove a conjecture published in 1989 and also partially address an open
problem announced at the Conference on Learning Theory (COLT) 2015. For an expected …

Whatever information a deep neural network has gleaned from training data is encoded in
its weights. How this information affects the response of the network to future data remains …

Despite their overwhelming capacity to overfit, deep learning architectures tend to
generalize relatively well to unseen data, allowing them to be deployed in practice …

While deep learning is successful in a number of applications, it is not yet well understood
theoretically. A theoretical characterization of deep learning should answer questions about …

We argue that the optimization plays a crucial role in generalization of deep learning models
through implicit regularization. We do this by demonstrating that generalization ability is not …

In this paper, we prove that depth with nonlinearity creates no bad local minima in a type of
arbitrarily deep ResNets with arbitrary nonlinear activation functions, in the sense that the …

Deep learning algorithms have been applied very successfully in recent years to a range of
problems out of reach for classical solution paradigms. Nevertheless, there is no completely …

This textbook establishes a theoretical framework for understanding deep learning models
of practical relevance. With an approach that borrows from theoretical physics, Roberts and …